any smooth affine hypersurface z of complex dimension n deformation retracts to a cell complex of real dimension n. starting from the newton polytope of the defining equation of z, i will give an explicit combinatorial construction of a compact space s, comprised of n-dimensional components, which embeds into z as a deformation retract. in particular, z is homotopy equivalent to s. the construction uses toric degenerations, nakayama-ogus's work in log geometry and the kato-nakayama space. it is motivated by the homological mirror symmetry program. if time permits, i will explain the connections. this work is joint with nicola sibilla, david treumann and eric zaslow.